I'm entertaining the idea of going with a synthetic deck as I'm a bit tired of the stain/paint/rot cycle of wood. I was a bit surprised to learn that most manufacturers are recommending that you install the planks onto a traditional wooden frame. My (potentially naïve) thought was that I'm not getting away from the stain/paint/rot cycle after all. Am I wrong? Or is there a use case here that makes it worthwhile (something about recycling the planks during a frame rebuild)?
The stain cycle is mostly related to appearance of the decking. Composite decking doesn't need stain.
The wood structure will still degrade but depending on your weather a pressure treated deck assembly can last a really long time with no maintenance. Wood stain never lasts a really long time, you are lucky to get more than a year out of it.
There are options that make a wood deck assembly last even longer. Minimally you can use joist tape. Other options are putting plywood down and then a torch on roofing membrane / flashing which is a membrane rated for a roof over a living area. This will certainly allow your wood structure to last 50+ years.
Holmes on Homes switched to steel deck structures - though that was during the time when the price of wood had skyrocketed to such a degree that the difference between the two wasn't that large.
If by recycle you mean re-use the composite planks that should be possible but the main advantage is you don't have to do any crazy maintenance for cosmetics and it looks great with just a sweep.
How long are you thinking a wood structure under a deck lasts? I would expect 15 years at a minimum and the stain interval is likely every 1-2 years or you accept it just looks on the terrible side and do it every 3-5?